In the introduction to 'A convenient setting for Global Analysis', Michor & Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open subsets of the modeling space for many modeling spaces."

But finite-dimensional manifolds are found to be interesting even though they can be embedded in some euclidean space (of larger dimension). (Actually this seems to me, to make the above claim intuitively plausible, so that claim should be no more than we should expect).

But they do go on to say that "Banach manifolds are not suitable for many questions of Global Analysis, as...a Banach Lie group acting effectively on a finite dimensional smooth manifold it must be finite dimensional itself.", which does seem a rather strong limitation.

In the introduction to 'A convenient setting for Global Analysis', Michor & Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open subsets of the modeling space for many modeling spaces."

But finite-dimensional manifolds are found to be interesting even though they can be embedded in some Euclidean space (of larger dimension). (Actually this seems to me, to make the above claim intuitively plausible, so that claim should be no more than we should expect).

But they do go on to say that "Banach manifolds are not suitable for many questions of Global Analysis, as  ... a Banach Lie group acting effectively on a finite dimensional smooth manifold it must be finite dimensional itself.", which does seem a rather strong limitation.